Linear Equation

A Linear Equation is an algebraic equation that follows a linear equation pattern (of the form [math]\displaystyle{ A \mathbf{x} = \mathbf{b} }[/math] or [math]\displaystyle{ a_1x_1+a_2x_2+...+a_nx_n=b }[/math] where [math]\displaystyle{ \mathbf{x} }[/math] is a vector of free numerical variables, and [math]\displaystyle{ A,b }[/math] are numerical coefficients).

Context:
- It can be a member of a Linear System.
- It can be interpreted geometrically as a straight line ([math]\displaystyle{ a_1x_1+a_2x_2=b }[/math]), a plane ([math]\displaystyle{ a_1x_1+a_2x_2+a_3x_3=b }[/math]), ...
Example(s):
- [math]\displaystyle{ 2x_1+3x_2=4 }[/math], a straight line passing through the axis points [math]\displaystyle{ (2,0) }[/math] and [math]\displaystyle{ (0,4/3) }[/math].
- [math]\displaystyle{ x_1+2x_2+3x_3=6 }[/math], a plane passing through the axis points [math]\displaystyle{ (6,0,0) }[/math], [math]\displaystyle{ (0,3,0) }[/math] and [math]\displaystyle{ (0,0,2) }[/math].
Counter-Example(s):
- a Non-Linear Equation, such as a polynomial equation.
- a Linear Inequality Equation, such as [math]\displaystyle{ A \mathbf{x} \leq \mathbf{b} }[/math].
See: Linear Scaling, Linear Constraint Equation, System of Linear Equations, Linear Algebra Concept.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/linear_equation Retrieved:2015-10-31.
- A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
  Linear equations can have one or more variables. Linear equations occur abundantly in most subareas of mathematics and especially in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state. Linear equations do not include exponents.
  This article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients and solutions in any field.

http://www.math.vanderbilt.edu/~msapir/msapir/jan10.shtml#oneequation
- QUOTE: A linear equation is an equation of the form : [math]\displaystyle{ a_1x_1+a_2x_2+...+a_nx_n=b \ (1) }[/math] where [math]\displaystyle{ x_1,...,x_n }[/math] are unknowns, [math]\displaystyle{ a_1,...,a_n,b }[/math] are coefficients.
  Example: [math]\displaystyle{ 3x-4y+5z=6 \ (2) }[/math] This equation has three unknowns and four coefficients [math]\displaystyle{ (3, -4, 5, 6) }[/math]. A solution of a linear equation (1) is a sequence of numbers [math]\displaystyle{ x_1,...,x_n }[/math] which make (1) a true equality.